Page - 47 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

The novelty of our approach is that anatomical a-priori information is learned from high-resolution CTdataandmodeledasaSSM,whichis thenfit to theangiogramsbya2-D/3-Dregistrationmethod. TheapplicationofSSMsfor recoveringshape fromangiographyhasbeensuccessfullydemonstrated byotherauthorsforhard-tissueobjects like thepelvis [4]or thevertebrae[1],butnotyet fornon-rigid contrast-enhancedsoft-tissueobjects liketheLV.Thispaper isarefinementofourpreviouswork[12]. For the sakeof comprehensibility, partsofSec.3 and4arebased thereon. 3. Methods 3.1. StatisticalShapeModels In order to build a 3-D SSM [2], a set of segmentations of the target shape is required. The contour of each shape Si is described by n landmarks, i.e. points of correspondence that match between shapes, and represented as a vector of coordinates: xi = (x1, ...,xn,y1, ...,yn,z1, ...,zn)i T. Allns shape vectors form a distribution in a 3n-dimensional space. This distribution is approximated by x= x¯+ Φb, with x¯= 1 ns ∑ns i=1xi being the mean shape vector and b being the shape parameter vector. By varying b, new instances of the shape class are generated. Φ is obtained by performing a principal component analysis (PCA) on the covariance matrixC= 1 ns−1 ∑ns i=1(xi− x¯)(xi− x¯)T. PCAyields theprincipalaxesof thisdistribution; theeigenvaluesgive thevariancesof thedata in the direction of the axes (= eigenvectors). To reduce noise and dimensionality only those eigenvectors withthelargestteigenvaluesareused. tdenotesthenumberofthemostsignificantmodesofvariation (MOV) and is chosen so that a fractionf of the total variation is retained, ∑t j=1λj≥f ∑ λj. Prior to statistical analysis, location, scale and rotational effects must be removed from the training shapes toobtainacompactmodel. Commonly,Procrustesanalysis is applied tominimizeD= ∑|xi− x¯|2, the sum ofsquareddistances (SSD)ofeachshape to themean. 3.2. ModelingofAnatomicalA-Priori Information ASiemensSomatomSensationCardiac64multi-sliceCTisusedtoacquire20datasetsat65%ofthe heartphase(R-Rpeaks)withaneffectiveslice thicknessof0.5mmandanaveragein-planeresolution of 0.33 mm. The size of the image mask in the transversal plane is 512×512 pixels; the number of slices varies between 220 and 310. The endocardial LV surface is manually segmented by experts in cardiology. Contours are specified in each fifth axial slice by interactively setting control points of a cardinal spline; intermediate contours are interpolated. The surface of an LV is represented as a stack of contours. Details like the atrial concavity, the apex and the aortic valve region are retained during segmentation to obtain an accurate model of the anatomy. Point correspondence among the training shapes is established based on back-propagation of the landmarks on a mean shape [11]. After segmentation, landmarkextractionandremoving location, scaleandrotationaleffects, theSSM isbuilt asoutlined in Sec. 3.1. Thefirst threeMOVof thefinalmodel are illustrated inFig.2. 3.3. LeftVentricularShapeRecovery In discrete tomography, a common strategy for solving the under-determined and ambiguous recon- struction problem is to use numeric optimization [3]. As an exact solution will usually not be avail- able, theprojectionsof the recoveredobjectneedonlybeapproximatelyequal to thegivenprojection data. In thiswork,a2-D/3-Dregistrationapproachisfollowedtominimizethedifferencebetweenthe given projections and the simulated projections derived from the SSM. To transform the SSM from 47